Use of optimisation in helicopter vibration control by structural

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PAPER Nr. : 68

USE OF OPTitHSATION

n;

HELICOPTER VIBR:\'flON CONTROL JlY STRCCTURr\L NODIFICATIO:i

by G. T. S. Done,

The City Uni ve rs i ;::; , Lend on ECl V OHB, England. H. A. 1.". Rangacharyulu,

Birla Inst. of Tech.

c,

Science, Pilani (Rajasthan), India ..

FIFTH EUROPEAN ROTORCRAFT AND POWERED LIFT AIRCRAFT FORUM

SEPTEMBER 4-7TH 1979- AMSTERDAM,THE NETHERLANDS

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USE OF OPTIHISATION IN IIELICO?ER VIBRATION CONTROL BY STRUCTURAL HODI?ICATION

by G.T.S. Done,

The City University, London ECl V OHB, England. H. A. V. Rangachar:;·ulu,

Birla Inst. of Tech. & Science, Pil&<i (Rajasthan), India. Abstract

The application of a mathegatical opt~mlsation process to heli-copter vibration control by structural modification is reported.

Attention is focussed on the reduction of vibration in the crew area using stiffness parameters as design variables. Use is made of forced vibration response circles to identify the parameters most effective in controlling the response in the crew area, thereby reducing the number of available design variables to a tractable size. The problem of reducing vibration is then cast as a non-linear programming problem and a sequential unconstrained minimization technique incorporating an:·- .. algorithm based on the methods of Davidon, Fletcher and Powell is used to determine the precise values of the parameters. The method is applied to a simple two-dimensional bean-element helicopter fuselage model, and the results discussed. A:though the model is too simple for useful deductions of practical signizicance to be made in the strictly engineering sense, the exercise does denonstrate what can and cannot be done in controlling vibration by using an optimisation routine.

l. Introduction

One of the many possibilities for vibration control of a heli-copter is to design the fuselage str~cture itself so that the vibration

response in the more important areas such as crew and passenger spaces

is minimised. Ideally this would ircvolve optimising the separate

ele-ments of the fuselage structure to achieve minimum response in the

desired area. It is usual to perfor3 structural analysis operations on an appropriate mathematical model, such as a finite element model of the structure comprised of many eleme~ts, but to treat each element as a variable for the purpose of optimisation wouid be beyond the scope of

the average optimisation computer prcgramme and could involve heavy computational costs. To avoid this, a subsidiary exercise is per-formed in which the best few elements that can be treated as variables

for reducing vibration are chosen. ~ne~ once the sensitive elements

are identified, a formal optimisatio= ccn be used to fix the precise

values of the parameters characteris~~g these elements.

In the last decade there has caen a considerable advance in the

area of structural optimisation unde~ dynamic constraints, as the

recent comprehensive reviews by Rao -l] and Venkayya [2] will testify. A portion of the research work that ~as been done is concerned with

optimisation of a structure under fo=~eC vibration, and this is

rele-vant in the present case. Sciarra 3

J

provides the apparently sole example of a specific application to a helicopter fuselage, although he

used an optimality criterion rather :~en a for~al mathematical

optimisa-tion procedure.. Parameter selectic::-_ before optimisation is not. covered

to any great extent in the literatur2; quite often, when the number of

parameters needs to be limited the c:-.oi ce is simply made initially on

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2~ Parameter selection

The identification of the best parameters for selection in the optimisation process is based on the way the response at a point behaves when a parameter is varied. If a structure is excited by a sinusoidal force, while either the mass at a point or the stiffness between two points (as represented by a spring) is continuously varied, then the response in the complex plane at some other point is seen to trace out a circular locus. The diameters of the response circles thus produced for each parameter can be taken as a simple yardstick for deciding which parameters are most effective in controlling the

response at the point under consideration; for exawple, a parameter

producing a large response circle diameter clearly has a relatively greater effect than one producing a small response circle diameter. From a listing of circle diameters in decreasing size order for dif-ferent parameters, the top few parameters are those selected for the subsequent optimisation process.

The circular response property was originally used by Vincent [4], and adopted and utilised in the way just described by Done and Hughes

[5,6]

and Done et al.

[7]

on two simple practical cases. 3. Response optimisation

The response to harmonic excitation of a general structure

modelled by finite elements and having n degrees o£ freedom is gLven by

R = G F (1)

G is the (nxn) complex receptance matrix bet<.;een all the points concer~d and is given by

G

=

(2)

where K, Nand Care stiffness, mass and damping matrices respectively and w Ts the citcular frequency of the exciting force. F and R are the column vectors (order (nxl)) denoting force and response

respectively.

If the structure is modified, say, by inserting linear springs 6f stiffness ki(i=l,m) between the nodes that have compatible degrees of freedom, the stiffness matrix K becomes a function of these variable stiffness parameters ki. Out of-all possible ki the best few are selected for optimisation by using the parameter selection programme. The particular responses ~p to be minimised are gLven by

R = S G F

-p (3)

where S is a sorting matrix that picks out the associated points. Now

in terffis of mathematical programming the problem is to minimise an

objective function which is a function of the elements of~' using the best parameters as design variables. In evaluating ~ the damping

matrix C is assumed to be zero, since for situations as in the present

case wh~re the forcing frequency is well removed from resonant

fre-quencies its contribution is negligible compared with those from mass

and stiffness. A major advantage is that complex natrices are thereby avoided. A further point is that the helicopter rotor speed is

sub-stantially constant, and therefore the excitation frequency is assumed constant.

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The optimisation is done by using a readily available variable matrix algorithm supplied by R.A.E., Farr.borough. The algorithm requires gradient information and is based on !:he methods of Davidon, Fletcher and Powell [8]; it is one of the most powerful gradient-based

algorithms and has a quadratic convergence property~ The gradients required can be obtained from differentiating eqn. (3):

oR

...::.!?.

ok.

~

oK

-

_{--ak. --}

S G-=- G F ~

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It can be seen from this that the receptance matrix G needs to be evaluated fer each recomputation of the gradient vector, thus neces-sitating the inversion of a (nxn) matrix (eqn. (2)) each time. The number of degt"ees of freedom n may be large and there is the possibility of using a lot of computer time. Hm.rever, an alternative expression

to eqn. (2) can be used to calculate the receptance matrix which is derived in Ref. 5 and reproduced in the Appendix:

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where

2.

0 is the receptance matrix of the unmodified structure i.e. with ki = 0.

The elements of the matrices A, B and D are formulated from the elements of the basic receptance mntrix-G_{0 •} -V is a diagonal matrix. of the variable parameters ki and I is an identity matrix. Here the matrices D and V are of the order (mx:m), 1.;here m i.s the number of vari-able parameters~ Any sensible and ~ractical modification would

involve relatively fe~v variable parar:ete:cs compared with the number of degrees of freedom, so m is much less than n and h~nce the size of the matrix to be inverted is only (m..'<rn). In the present \Wrk the

gradients are computed directly from the e::..'"Pression resulting from differentiating eqn. (5).

4. Experience on a simple gear box/engine/fuselage illodel

As a test case for the application of the optimisation method to dynamic problems, a simple system representing a ·helicopter in which the gear box, engine and fuselage are replaced by rigid bodies as shown in Fig~ l is coP.sidered. The gear box and engine mountings are

treated as variable stiffness parameters. The n;odel has eight C.egrees of freedom and six variable spring stiffuesses~ The gear box is sub-jected to an oscillatory forcing mome:1t cnC the a:.::t is to find Hhat variable stiffness values ki are necessary to r.:ake. the vibration at the pilot's seat, P, zero or as low as possibie~ lhe stiffnesses of the springs are given by (kb)i + ki (i=l,5), wt:e.re (kb) are the ba.sic stiffness values.. The stiffnesses k and k effectively <ict as one

l 2

stiffness.

Realistic bounds on the stiffness values k:, are basic objective function to be minio.ised is ta..l(en as

C

is the vertical response at P~

imposed and a ., 2

-· h-p, vhere Rp

Now with a simple objective ft:...'l.c~icn of this type i t must be realised that a unique opti111um does not exist. For example ·with only two variables k₃ and k₄ (the gradiencs i .. ~ith respect to k

1, k2, k5 and k₆ were very small anci thus these variables are no-:: i2portant in the optimisation.) zero vibi"ation respons~ is obtaincC. ;.;ften k:.; == f(k₃) in

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\.,

__

Stiffness parameter ki

=

(ki - kie)/(kiu- kie) ~here kiu, kie are upper, lower bounds on i-th stiffness (i.e. 0 ~ ki ~ 1).

Fig. 1. Gear Box/Engine/Fuselage Model.

Even in the presence of damping i t can be sho••n by using response

cir-cles that there exist two optimum solutions for this case. For small

damping the shape of the objective function ¢ is such that the response is second order small along k,

=

f(k

3) which makes it hard to find the

minima numerically and the situation can be worse ior the case of many

variables. Thus there is a need to choose the best solution out of all possible solutions. The simplest means in the present case is to

assume that an increase in stiffness normally i=plies an increase Ln

weight and to seek the solution which provides the least '"eight penalty, i.e. the least total added stiffness.

Thus the problem can be recast as a constrained problem,. VlZ.

Hinimise l:ki, subject to the cons trairct

P

= ~l_g_~p = 0

where

g

is a matrix which allows a quadratic function of response

mag-nitudes at various points to be formed and Eki is the sum of added

stiffnesses. This opti1msation algorithm is used sequentially in the present problem on a modified objective function defined by

~

= l:k. +

S¢

2 (6)

ens:.:re the minimum

possible response within the stiffness (or '..reight) oounds allowed. The nature of the result obtained for the t~J variable case is

shown in Fig .. 2. This represents a case where zero response can be

obtained within the range of variation of k₃ and k0 considered. The optimisation process has been started arbitrarily at k₃ = k₄ = 0.1 and

it is clear from study of the Figure that the miLic~w value of ~ occurs

roughly at the intersection of k3 = 0 and Rp = f(k3,k4) = 0. For an

initial low value of

S

the solution moves to point 2; thereafter,

S

is increased sequentially (steepening the slope of t"-e ?enalty function

surface) and the solution moves to the final one, satisfactory convergence being achieved after 5 steps.

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Fi"nisA oFr.er

5"

sfeps

Penc..l!y Fn

Ba.s-/c o E:f

f'n.

St.Jr/'qce

(0· Ofj)

;/f/~r4---/.~

-2.

f:/"

0 Sfqrf { 3

·6

9)

..._ A

=

o·t

'f

Z!ero

respon!:.f!'

ctf

P cdonJ

fAis

ftne

Rr

~

f{k

3 ,

~<'

11 .)"'

o

Fig. 2.

Visualisation of Optimisation.

The model considered here is very simple and to assess how this method ;rorks for a more complex model, a two-dimensional beam element model ("stick" model) of the Westland Lynx helicopter is studied. The details of the model and the results of application are presented in the next section.

5. Application of Lynx "stick" model

The aforementioned optimisation technique using the modified objective fu~:ction is now applied to the problem of minimising the vibrational response in the region of the pilot's seat of the Westland Lynx helicopter. The model used is show-n in Fig. 3. This "stick" model has 25 tapered beam elements and 60 degrees of fr<~edom with two

translational and one rotational at each node. The excitation on the structure is 2-'l oscillatory couple of frequency 21.7 Hz applied to the

rotor head (node 8) as shown in the Figure. Only stiffness changes corresponding to the adjacent nodes are considered, and the vertical response at the pilot' a seat (node 18) denoted by R₅₃is minimise-4~.

The suffix "53" refers to the degree of freedom number- its relation to structure node number is found by reference to Fig. 3.

The first twenty parameters obtained from a response circle

dia-meter listing are given in Table 1 and the par~ters are indicated by

the respective degree of freedom numbers. A li!odal damping'· ·facta:t :of· 2% critical is assumed for getting the parameter listing. The para-meters are also indicated in order of importance in Fig. 4 and are

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::z:

fl.o.f'. eoord.

-y.s""'<:.,..

[) X

:f

IS""

Degree of freedom co-ordinates for node i are (3i-2,3i-l,3i). Thus, e.g. co-ordinates for node 18 are (52,53,54) and for

node 2 (4,5,6).

Fig. 3. Lynx "Stick" Model - Node & d. o. f. Numbering.

TABLE 1

Parameter listing for the response R₅₃based on circle diameters

Parameter _Parameter Corresponding Unmodified _stiffness

identifying

co-ordinates Direction structural value ki

no. nodes (N/m) l 25 28 X 3-10 0.398620 X 107 2 7 10 X 3- I. _0.398620 _X ₁₀7 ~ 3 4 l3

z

2- 5

o.

398620 X 107 4 14 17

z

5- 6 0.525380 X 10 8 5 5 14

4-19

i

0.210152 8 20 2 53

z

1-18 1 o.216385 X 10 68-6

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Ro

Fig.

4. Stiffness Parameters (Identification Numbers).

The optimisadou algorithm requires the lo'Cler and upper limits on the variables. Two types of bound are chosen, firstly, all the stiffness parameters are each allowed a ~~iform variation of 0-1.75 x 109 N/m (0-107 lbf/in) and, secondly, a .range of variation of 20% of the basic stiffness values is also taken. For each set of bounds dif-ferent response cases are considered although the listing of best parameters is not strictly applicable to response cases other than

tlm•

'

The basic objective functions ~i corresponding to the different response cases (see Fig. 5) considered are

where the suffices on the R's refer to degree of freedom numbers. Hinimising a multiple response involving more than one node as in the case of ~3 and ~

4

can be thought of as reducing the vibratory response over a region of the structure. These response cases are illustrated in Fig. 5.

The computer results for the different cases of response and stiffness parameter constraints follow in Tables 2 to 9 inclusive.

For ea~h case the response function ~i used is staced, and also type of

upper bound nki on the additional stiffness variations nki. This max

is either simply 1.75 x 109 N/m throughout or 0.2ki, where ki is the existing stiffness paremeter value of the unQOdified structure. A

typical value is obtained from the stiff~ess matrix; for example, a stiffness connecting points p and q on the structure is given by the negative of the pq-th element in the matrix. The lower bounds on nki are all zero.

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~

,._-.~f-~,._

_

_,y/

Fig. 5.

Response Cases Considered.

The magnitude of the response or composite response

C~i)~

is given as an acceleration for a nominal excitation pitching moment at the rotor head of 5650 N m.

The parameters listed in Table l are used for a sequence of runs

in which the first two, four, six, ten etc. parameters are used in the

optimisation, the optimum values being shown. The response is given as a ratio of that for the unmodified structure, and the sum of the additional stiffnesses indicated. Also given are the C.P.U. times. The parameter identifying number appearing in the first column of Tables 2-9 is that which appears in the first column of Table l and also in Fig. 4.

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0' 00 I

"'

TAULE 2 Cnse 1 Par.o.m. ident. no. 1 2 J 4 5 6 I 8 9 10 II 12 13 14 15 16 17 18 19 20 Response function ~l a R; 3 Sti(fneoa bounds 0 ~ bk. $ 1.75 x 109 N/rn l Rl!sponne m.lgaitude (tJ

1

)~-=} 0.132g

Opt. values of 6ki/6ki max

for cliff. nos. of params.

2 4 6 10 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1936 0.0059 0.0059 0.0059 0.0161 0.0161 0.0161 () () () ~-···~--0 0 0 0 0 0 0 0 0 0 0 0 0 ---·--::-

-f.(P.'<.xlo-7 0 0.1936 0.022 0.022 0.022 1 -&!sp. ratio CPU

I

{~J

time _i (min) 20 0 0 1.0 0.025 0 0 0 3.5 Q 0 () _3.2 ----~~ --~---0 ' 0 0 0 0 4.0 0 0 0 0 0 0 15 _I 0 0

_i

0 I 0 i o. 0037 . 0

I

45

I

0.0037 TABLE 3 Case 2 Par am. ident. no. I 2 3 4 5 6 7 B 9 10 11 12 13 !4 15 Response function ~

2

• a;₃+ _{R; 2}

Stiffness bounds 0 .t; hki (. 1. 75 x 109 N/lo

Response magnitude (~

2

)~ ~)Q,I,OJg

Opt. values of 6ki/6ki

tn3X

for diff. nos, of pararns.

2 4 6 10 15 0 1.0 0 0 0 0 1.0 0 0 0 1.0 0 0 0 1.0 1.0 !.0 0 0 1. () 0 0 () ₀

---···

1.0 () 0 0 1.0 1.0 !.0 0 0 1.0 1.0 !.0 1.0. E(6k.xlD- 7) 0 4.0 1.0 5.0 5.0 1 Renp. ratio CPU

{(.:d

tin~ (ruin) !.0 _--~1 o. 322

___ d

(l, f\70 l.O _,. ... " _·-·

-,

0.939 1.7 i

I

0.011

I

~-~

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"'

cc I >-' 0 TAELE 4 Case J -l'arnm, ident. no. 1 2 ) 4 5 6 7 8 9 10 11 12 13 14 15 f . J 2 2 Response uuct10n ¥' 3 = R53 + R5 Stiffness bounds 0 ~ 6k. ~ 1. 75 x 109 N/m

'

Response magnitude (~

3

)! =}0.190g

-Opt. vnlues of bki/liki

max

for cliff. nos. of params,

2 4 6 10 15

-0 1.0 0 0 0 0 l.O 0 0 0 ~ --·--·· l.O () 0 0 0.0173 1.0 0.3~70 0. 3 7 57 -1.0 0.2374 0.2545 0.0037 0.0035 0.0036 0 0 0. 7790 0.7467 0 0 0 0 · - · - · 0 0 0 0 0 E(6k. x1o-7) _l 0 3.0173 2.0037 1. 3769 1. 3805 -RPsp. ratio CPU

{(.:h1

(min) timl! 1.0 0.025

---·--

---o. 818 0. 5 0 1.7 0 5 . .

-.

0 57 ' i TABLE 5 Case 4 Par am. ident. no. 1 2 3 4 5 6 • 7 8 9 10 11 12 13 14 15 • J 2 2 2 2 Response funct1on ¥1 4 = R53 + R52 + R5 + R1, Stiffness bounds 0 ~ 6k. ~ 1.75 x 109 N/m

'

Response magnitude (~

4

)! =4 0.57lg Opt. values of bk./6k. 1 ' mnx

for diff. nos. of params,

4 6 10 15 0 0 0 0 0 0 0 0 0 u 0 0 1.0 1.0 1.0 0 o.ooot. o. 0008 0 0 0 0 0 0 0 0 0 1.0 1.0 0

---0 1.0 1.0 1.0 J.O E(6k.x10- 7) 1.0 l 1. 0004 1. 0008 5.0

----Resp. ratio

{"!

_a4~o}

0.868 · -a. 867 ---·-0.863

--·---0.011 CPU time (1ni n) o. 7 1.1 '~. 5 1'

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"'

I >-' >-' TABLE 0 ~ rar:tm, i dent. no, I 2 3 4 5 & 7 8 9 lO ll 12 l3 14 Response function 4 1 .. R~

3

Stiffnes5 bounds 0 ~ ~ki ' 0.2ki Response magnitude ("

1

)!=---}

0.132g

Opt. values of 6ki/6ki

ma:x

for diff. nos. of params.

2 4 & 10 0 0 0 0 0 0 0 0 ---· 0 0 0 I l I I I 0 0 --·- _I l I 0

---·--I 15 0 0 0 I I 0 I 0 I 0 I I I 1 I

c~o-it

o o.oo6o 0.0089 0.0276

j

202.7

----

· ' -[(~

'---I

Resp. ratio CPU

{(~:~j

time (min) 1.0 0.027 o. 781 o. zt. 0.538 0. 34 0.537 1. 24 -·--:--

--·---' 0.002 33 TABLE 7 Case 6 Par am. ident. no. 1 2 3 4 5 & 7 8 9 10 II 12 13 14 15 Response f\Lnction r\ 2 "' R;) + F.~z Stiffness bounds 0 ~ 6ki ~ 0.2ki

Response magnitude (~

2

)!

=>

o.t.o3 g

Opt. values of bk./bY .. l l

max

for cliff. nos. of paraws.

-2 4 6 IO 0 I 0 0 0 I 0 0 I 0 0 0 I I · -I I 0 0

- - -

_I 0 I I

---

-

--

- · /.(~l<.i=<l0-7) 0 0.0137 0.0089 3.202

-ResF rati 15 { .2

<'iiT

-0 0 1.0

---0 1 0.99 ---I 0 0,91,1

----0 1 1 0 o. 93! ·-··· 1 1 1 I I 0.00 202.7

---CPU li WI' (min) 0.033 0.067 0. 15 o. ~5

-1-~

--- .

_L

__

_j

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"'

9'

...

N TABLE 8 ~ Par am. idcnt. no. 1 2 3 4 5 6 7 6 9 IO I I 12 13 14 15

--Response function ~J •

R; 3

+

R;

Stiffness bounds 0 ~ 6ki ~ 0.2ki

Response magnitude (lj: )I ~0.190g

3 0

~1t. v9lues of 6k

1/Aki _mox for diff, nos, of params.

2 4 6 10 0 I 0 0 0 1 0 0

---

₁ 0 0 0 1 1 1 1 0 0 1 1 1 ()

____

..

r.(6k.x10-7) 0 0.001/1 0.0089 0.0276 l Resp. ratio CPU { • ! tiroe

(J3~o}

(min) 15 0 0 _:.:_o ___ 0.033 0 1 0.994 0.12 1 0 0.573 0.33 1 1 I 0

o.

5 72 o. 2 7

---

---I I _' I I I 0.001 26

_I

L__j

202,7 ---TABLE 9 ~ Par am. ident. no. 1 2 3 4 5 6 7 6 9 10 II 12 13 7.4 15 . 2 2 2 2 Response function C₄a RSJ + R52 + RS + R 4 Stiffness bounds 0 ' 6k. ' 0.2k. _l _l

Response magnitude (~,)I...,.>0.57lg

" 0

Opt. values of Ak./Ak. l l

max

!or diff. nos, o( params.

4 6 10 0 0 0 0 0 0 0 0 0 0 I 1 I 1 0 0 I I 1 I

---15 0 0 0 I I 0 0 I I 0 I I I I J f.(6k;xl0-7) 0 o.0069 3. 202 202,7

-Resp. ratiCI CPU

{«:~0

1

(min) t-ime --'-1.0 0.067 o. 945 0. I 5

---

· -0.9JB ().{d! ---·

---

'"-~·-- -0.001 25 . - - - - . ----

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-6. Discussion and ~onclusions 6.1 The optimisation process

The problem of minimising tioz ·:ibration on the Lynx model has been formulated so that a straightfor~ard optimisation process can be applied. In doing this, many pracci2al aspects have had to be

ignored, but discussion of this is presented in the following section.

The optimisation exercise hcs ::e:en extre:::;ely useful in that an

idea of the limitations, and hence its future usefulness can be formed.

The most important limitation is that the process itself, or rather the

particular process adopted, is very e:.:::t.ravagan.t of computer time.

This has been found to vary approxiDntely with the cube of the number

of variables considered, thereby i~osing a prac~ical top limit of

about 20 variables.. This represented a serious drawback because the

present exercise was essentially exploratory ~cd the time left for development was thus drastically curtailed.

Before discussing further drawhacks, it is instructive to

examine individually each case studied. In Case l (Table 2), zero response was obtainable and the optimisation proceeded without

compli-cations. Parameters 4 and 5 are seen. to provide all that is necessary,

until the final introduction of para::::-.e::e:r- 20 w~ich gives a solution of

lesser total added stiffness (discussion of the added weight is

reserved for the following section). Czse 2 il2.ustrates that a problem

of finding a global maximum has occ~rr2d. This case was run several

times with different input conditions 2ad for the 10 and 15 parameter

cases, a variety of solutions was obca~ned. In the event, a low

response solution was found (15 parazecers), but the results taken together do not represent a sensible a~d ordered sequence. Case 3, on the other hand, shows a successful seq~ence, as does Case 4, although the latter indicates a rather heavy pe~alcy (in added stiffness) to pay for the reduced response. The remaining Cases 5 to 8 may be

con-sidered together. In these the stiffr:ess bounds were taken to be 20%

of the existing stiffnesses, and throusho~t, the variable sti££nesses

tvent to either their upper or lower li-,its. It is interesting that r.n

these cases, regardless of the respo~se function taken~ the final

stiffness configuration obtained is ~o~ghly the saQ.e.

From this study of the separace cases it ·cay be seen that a second important drawback is the inaoility in Case 2 to find global

m1n1.ma. When it is suspected or kncT,..•n_ t!:at a global minimum has not

been found, then the case under inves ti g-,at.ion nus t. be re-run with

dif-ferent starting values of the parameters a:::td of r:re penalty function coefficient

P•

Thus, the human ope::ac:e;r enteoc::s into the picture and judgement, backed up by experience, ::usc be used to obtain a better

answer. In cases where a zero or near-zer-o response. \-Tas obtainable, finding the global minimum did not see:-_ to be 2 p:-oblem. Hmvever, in

the opposite situation, there was a te:::"ency for all parameters to adopt either their upper or lower li=its, and, particularly in Case 2,

a number of different minima were fow1C.. To sor::-e extent, the par-ticular minimum found is pre-determi~eC ac the first and lowest value

of

6;

thereafter, for increasing val~es o£

8

the response part of the objective function dominates at the e:<?22Se of the added weight part.

Finally, there must remain the c;.·.:estion of whether or not the

optimisation process used here was the ~ight one. Time did not allow

the development of other methods.

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6.2 The use of optimisation in practical proble=s

It is clear from Tables 2 to 9 that using the response circle diameters for initially selecting the best paraceters for optimisation is not altogether satisfactory. Ideally, the para3eters that turned out to be important in the optimisation should have appeared at the top of the circle listing. However, the circle diameter method as used

here is very simple, and a more realistic criterion, still based on

circle response properties, can easily be employed. This is something that warrants further investigation.

The gear box-engine-fuselage model of Fig. l was simply used to test out and gain experience with the optimisation routine. The "stick" model of the Lynx provided, however, scope for a more serious attempt at structural optimisation. Even here, though, the model was not sufficiently representative for useful practical deductions to be made. The major drawback was the absence of a relationship between element weight and element stiffness. Future models, even if simpli-fied by restricting the number of degrees of freedom, should have the element weights involved in a proper Eanner.

Unlike flutter optimisation for fixed wing aircraft, the basic objective function is somewhat arbitrary. Human judgement is required to formulate a suitable response function for embodiment in the objec-tive function. Clearly, it is impossible to reduce the response in the fuselage everywhere to zero, so it has to be decided where the response should be small, or zero, and in which directions. If a com-posite response is to be used, some sort of weighti2g has to be decided

upon for the various components~

A further practical consideration is that the optimisation should be terminated when the payoff between added ;reight and response reduction becomes unacceptable. Such a bounding relationship could be provided by information obtained from industry.

7. Acknowledgements

The authors wish to thank many people at westland Helicopters Limited and also at the Royal Aircraft: Establishment, Farnborough, for their continued help, support and encouragement.

,8. References

[1] S.S. Rao, Structural optimization under shoe~ and vibration environment, The Shock and Vibration Diges~,

Q.

(2) Feb. 1979.

[2]

[ 3]

[4]

V.B. Venkayya, Structural optirization: recommendations, Int. J. Num. Hethods 1978.

A reviet>~ and some

in E:lg. ,

Q,

203-228,

J.J. Sciarra, Use of finite eleillBnt damped forced response

strain energy distribution for vibration reduction, Boeing

Vertol Rep. D210-l0819-l, July 1974.

A.H. Vincent, A note on the properties of the variation of

structural response with respect to a single structural

parameter when plotted in the complex plane, Westland Helicopters Rep. GEN/DYN/RES/OlOR, 1973.

(16)

ls]

G.T.S. Done and A.D. Hughes, The response of a vibrating

structure as a function of s~ructural parameters, J. Sound &

Vib., 2§_, 255-266, 1975.

[6] G.T.S. Done and A.D. Hughes, Reducing vibration by structural modification, Vertica,

l•

31-38, 1976.

1

·.7]

G.T.S. Done, A.D. Hughes and J. Webby, The response of a

vibrating structure as a functiorr o£ structural para~eters

Application and experiment, J. Sound & Vib., 49 (2) 149-159, 19 76.

[s]

D.A. Pierre, Optimization cheory with applications, John Wiley, N.Y., .1969.

Appendix

Matrix express~on for the modified receptance

A brief derivation of eqn. (5) of Section 3 is g~ven here.

Consider a structure modified by inserting two linear springs of rates

k₁ and k₂ betwee!l points a and b, ap~d c and d respectively, and excited

by a harmonic force Fp at p. The o:Ojective is to obtain !:he recep-tance Gpq between point p and a further point q when the structure is thus modified.

The forces exerted by the springs Fa, Fb and F c, F d at the points of attachment a, b and c, d can be written as

F

a R ) a

where R denotes the displacement res?onse.

The response

Rq

can now be written as

R q

where Gij is the recep tance be tween the points i a!1d J ~

Using eqns. (Al) this can be re-written as

(Al)

(A2)

where

o

₁

= Rb -

Ra and

o

₂

=

Rct -

Rc

are the relative displacew~nts of the springs.

(17)

Similar expressions for the displacements at a, b, c and d can also be formed: R = G F + (G _{- Gab)kl}

0

_l ₊ (G _{- Gad)k2o2} a ap P a a ac

~

Gb F _{p p} + (G _ba- cbb)kl

01

+ (G _be- Gbdlkz6z R = G F + (G _{- Gcb)kl}

₀₁

₊ (G _{- Gcd)kzoz} c cp P ca cc Rd Gd F _{p p} + (G _da- Gdb)kl

0

l + (Gdc - Gdd)k2°2 and by combining these the relative displacements <5

1 and <52 can be expressed

where

o=FB-DVo

p-Rewriting eqn. (A2) as R=GF+AVo q qp p (Gad - Gac)l (G d - G ) c cc (A3) (A4)

where A= (Gga- Gqb Gqc- Gqd),

i

can now be eliminated using eqn. (A3) to prov1de the new receptance Gnew between p and q, i.e.

G = R /F new q p

G +

!::_~

(I_

+

~~)

-1! qp

The expression can be generalised to include

and !::_,

!

and~ can be easily formulated from

the unmodified structure. The matrix to be as the number of variable spring stiffnesses

68-16

(AS)

any number of variables the receptance matrix of inverted is only as large considered.